The Rossoneri recorded a scoreless draw in the first leg at San Siro last week, and appeared to be on their way to the next round when Antonio Nocerino made it 1-1 at Camp Nou before the break.
However, Barcelona netted two more goals in the remainder of the game to record a 3-1 win and eliminate the Serie A champions.
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"We fought hard and made them suffer. We really scared them because with a draw we would have gone through," Abbiati was quoted as saying on the club's official website.
"It's a shame how we conceded the third goal. We go out with our heads held high, but there is bitterness because after the first-leg draw we believed in ourselves. It’s a real shame."
Abbiati then looked to the remainder of the season and stressed that AC Milan will now have to turn their attention to Serie A again.
"In the league we have eight more finals to play. We have a two-point lead and have to hold onto it tight. From tomorrow on, we’ll be focused on the league again as it’s the only objective left."
AC Milan resume Serie A action on Saturday when they take on Fiorentina.